OILER Secondary School

Numbers, operations, mental arithmetic

PITAGORAS is a didactic tool aimed at secondary school that focuses on mental calculation. We believe that developing the ability and consequently the habit of mental calculation not only helps whenever you need to calculate something, even in written form, but in general helps acquire the sense of number. Mental calculation is therefore useful both in didactic environments, for example to check the result if the calculation was performed in another way or to estimate an order of magnitude, and in everyday life, on all occasions when quantitative reasoning is required. PITAGORAS uses the completion of a “pyramid” as its main technique: we will explain how this works in detail later, anticipating for now that each brick contains a number and that you go from one level to the upper one by performing an operation. In certain contexts, this structure can be funnier and richer than usual computation, as we will see later. In particular, it implies the idea of inverse operation and can be useful as an implicit introduction to the concept of an equation.
cards

Data Sheet

CLASSES: 6, 7 and 8.

Skills:
The pupil:

  • calculates confidently, also with rational numbers, masters their different representations and estimates the size of a number and the result of operations;
  • uses and interprets mathematical language (Cartesian plane, formulas, equations, ...) and grasps its relationship with natural language.

METHODS:
The pathway starts from activities on addition and multiplication to promote mental calculation.
Several studies stress not only the importance of mental calculation for understanding arithmetic, but also the need to move from an additive to a multiplicative scheme for a better conceptualisation of number (Maclellan, 2001). The teacher will therefore guide students along a path that requires growing skills and competences. Students should get used to calculating in their heads without writing anything down, because the strategies used on paper differ from those of mental calculation. On the other hand, some algebraic properties turn out to be useful strategies in mental calculation.

This pathway was designed by Luigi Bernardi and Giorgia Damiano.

No activities yet: more are coming soon!

PITAGORAS is a didactic tool aimed at secondary school that focuses on mental calculation. We believe that developing the ability and consequently the habit of mental calculation not only helps whenever you need to calculate something, even in written form, but in general helps acquire the sense of number. Mental calculation is therefore useful both in didactic environments, for example to check the result if the calculation was performed in another way or to estimate an order of magnitude, and in everyday life, on all occasions when quantitative reasoning is required. PITAGORAS uses the completion of a “pyramid” as its main technique: we will explain how this works in detail later, anticipating for now that each brick contains a number and that you go from one level to the upper one by performing an operation. In certain contexts, this structure can be funnier and richer than usual computation, as we will see later. In particular, it implies the idea of inverse operation and can be useful as an implicit introduction to the concept of an equation.
cards

No activities yet: more are coming soon!

Data Sheet

CLASSES: 6, 7 and 8.

Skills:
The pupil:

  • calculates confidently, also with rational numbers, masters their different representations and estimates the size of a number and the result of operations;
  • uses and interprets mathematical language (Cartesian plane, formulas, equations, ...) and grasps its relationship with natural language.

METHODS:
The pathway starts from activities on addition and multiplication to promote mental calculation.
Several studies stress not only the importance of mental calculation for understanding arithmetic, but also the need to move from an additive to a multiplicative scheme for a better conceptualisation of number (Maclellan, 2001). The teacher will therefore guide students along a path that requires growing skills and competences. Students should get used to calculating in their heads without writing anything down, because the strategies used on paper differ from those of mental calculation. On the other hand, some algebraic properties turn out to be useful strategies in mental calculation.

This pathway was designed by Luigi Bernardi and Giorgia Damiano.

Pitagora of Samos

The stories about the life of Pythagoras are a mix of legend and reality. Nothing is certain, because he left no writings, but a good deal of information has come down to us through his disciples and some writings of Aristotle, Herodotus and Plato. So we know that he lived in the 6th century BC, that he was born in Greece but travelled widely in Asia Minor and Egypt (it is said that it was Thales who introduced him to the mathematical knowledge of the East), before reaching Italy, at Croton, where he founded a School: a scientific, philosophical and religious community whose disciples followed certain moral rules and pledged not to divulge the philosophical and mathematical knowledge they had learned from their master Pythagoras.
Everyone knows the Pythagorean theorem, but in fact the Pythagorean doctrine was based not on geometry but on arithmetic. The other disciplines that made up mathematics, namely music, geometry and astronomy, were founded on arithmetic. For Pythagoras, number was the essence of all things. Knowing the properties of whole numbers was part of a mystical path, and special representations, such as triangular or pyramidal numbers, took on great importance.
It seems that the Pythagorean school was thrown into crisis by the discovery of irrational numbers, or rather by the fact that the ratio between the diagonal and the side of a square cannot be expressed as a ratio between whole numbers. But that is another story...